Orthogonal rational functions on the real half line with poles in [-∞,0]

The main objective is to generalize previous results obtained for orthogonal Laurent polynomials and their application in the context of Stieltjes moment problems to the multipoint case. The measure of orthogonality is supposed to have support on [0,∞) while the orthogonal rational functions will have poles that are assumed to be “in the neighborhood of 0 and ∞”. In this way orthogonal Laurent polynomials will be a special case obtained when all the poles are at 0 and ∞. We shall introduce the restrictions on the measure and the locations of the poles gradually and derive recurrence relations, Christoffel-Darboux relations, and the solution of the rational Stieltjes moment problem under appropriate conditions.